On the Multiplicity of a Proportionally Modular Numerical Semigroup. (8th September 2021) - Gu, Ze
Abstract : A proportionally modular numerical semigroup is the set S a, b, c of nonnegative integer solutions to a Diophantine inequality of the form a x   mod   b ≤ c x, where a, b, and c are positive integers. A formula for the multiplicity of S a, b, c, that is, m S a, b, c = k b / a for some positive integer k, is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k . Furthermore, we obtain k = 1 for various cases and a formula for the number of the triples a, b, c such that k ≠ 1 when the number a − c is fixed.